| | Quadratic | Linear with kernel weight |
| (Intercept) | 0.769*** | 0.819*** |
| (0.034) | (0.015) |
| Income_Centered | -11.567 | -23.697*** |
| (8.101) | (3.219) |
| Participation | 0.093** | 0.033 |
| (0.044) | (0.021) |
| I(Income_Centered^2) | 562.247 | |
| (401.982) | |
| Income_Centered × Participation | 19.300* | 26.594*** |
| (10.322) | (4.433) |
| Participation × I(Income_Centered^2) | -101.103 | |
| (502.789) | |
| * p < 0.1, ** p < 0.05, *** p < 0.01 |
]
---
# Fuzzy regression discontinuity
<br>
.vcenter[
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_c5ea377e283f_files/figure-html/unnamed-chunk-12-1.png" alt="10: A causal diagram that fuzzy regression discontinuity works for" width="60%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">10: A causal diagram that fuzzy regression discontinuity works for</p>
</div>
]
---
# Fuzzy regression discontinuity
<br>
**Sharp RDD vs. FRDD**
<br>
- In FRDD, the data cannot simply be limited to the area around the cutoff to control for `\(\text{Running Variable}\)`
→ Doing that would lead us to understate the effect!
- Instead we apply IV:
- The first stage uses `\(\text{AboveCutoff}\)` as an instrument for `\(\text{Treated}\)` (as well as `\(\text{Interactions}\)` )
- Estimate regression discontinuity equations as for the sharp RDD in the second-stage equation
---
# Fuzzy regression discontinuity
<br>
**IV estimation of FRDD**
<br>
IV divides the effect of the instrument on the outcome by the effect of the instrument on the endogenous/treatment variable.
→ The effect of being *above* the cutoff on the outcome is scaled but divided to account for the fact that being above the cutoff only leads to a partial increase in treatment rates.
---
# Fuzzy regression discontinuity
.vcenter[
.blockquote[
### Example: Effect of mortgage subsidies on home ownership (Fetter 2013).fn[3]
- Fetter’s main research question is how much of the increase in the home ownership rate in the mid-century US was due to mortgage subsidies given out by the government
- He considers people who were about the right age to be veterans of major wars like WWII or the Korean war:
Anyone who was a veteran of these wars received special mortgage subsidies.
]]
.footnote[[3] Fetter, Daniel K. 2013. *How Do Mortgage Subsidies Affect Home Ownership? Evidence from the Mid-Century GI Bills*. American Economic Journal: Economic Policy 5 (2): 111–47.]
---
# Fuzzy regression discontinuity
.vcenter[
.blockquote[
### Example: Effect of mortgage subsidies on home ownership (Fetter 2013)
- There is an age requirement to join the military:
If one is born one year too late to join the military to fight in the Korean war, then he will not get these mortgage subsidies (or at least far fewer veterans were eligible).
→ Discontinuity based on birth year.
- The “treatment” of being eligible for mortgage subsidies would only apply to some people born at the right time:
Treatment rates jump from 0% to some value below 100% (fuzzy).
- Veteran status at this margin increases home ownership rates by 17%
]]
---
# Fuzzy regression discontinuity
</br>
.blockquote[
### Example: Effect of mortgage subsidies on home ownership (Fetter 2013)
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_c5ea377e283f_files/figure-html/unnamed-chunk-13-1.png" alt="11: Eligibility for mortgage subsidies for being a Korean war veteran and home ownership from Fetter (2013)" width="70%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">11: Eligibility for mortgage subsidies for being a Korean war veteran and home ownership from Fetter (2013)</p>
</div>
</br>
]
---
# RDD
<br>
**Placebo tests**
<br>
- The astonishing thing about regression discontinuity is that it closes all back doors, even the ones that go through variables which cannot be measured
- That is the whole idea:
Isolate variation in such a narrow window of the running variable so that it is plausible to claim that the *only* thing changing at the cutoff is treatment—and by extension anything that treatment affects (like the outcome)!
---
# Placebo tests
</br>
**Idea**
Anything we would normally use as a control variable should not affect treatment.
<br>
**Procedure**
- Run the regression discontinuity model on plausible control variables
- If an effect is found, the original RDD might not have been right. This might indicate that our assumption about randomness at the cutoff is violated.
---
# Placebo tests
</br>
**Procedure**
<br>
Keep in mind that, since we can run placebo tests on a long list of potential placebo outcomes, it is likely that we find a few nonzero effects just by random chance.
→ If one test a long list of variables and find a few differences, that is not a fatal problem with the design.
In these cases, it is better to add the variables with the failed placebo tests to the model as control variables.
---
# Placebo tests
.vcenter[
.blockquote[
### Example: Manacorda et al. (2011)
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_c5ea377e283f_files/figure-html/unnamed-chunk-14-1.png" alt="12: Performing regression discontinuity with controls as outcomes for a placebo test" width="70%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">12: Performing regression discontinuity with controls as outcomes for a placebo test</p>
</div>
]]
---
# The density discontinuity test
</br>
**Random assignment may fail**
</br>
There are two ways manipulation could happen:
- First, whoever (or whatever) is in charge of setting the cutoff value might do so with the knowledge of exactly who it will lead to getting treated.
- Second, individuals themselves likely have some control over their running variable. Sometimes they have *direct control* and sometimes they have *indirect control*.
---
# The density discontinuity test
</br>
**Random assignment may fail**
</br>
In the case of indirect control we do have a test we can perform to check whether manipulation seems to be occurring at the cutoff.
For this we inspect the distribution of the running variable around the cutoff:
- If the running variable was randomly assigned without regard for the cutoff we expect its distribution to be smooth
- A distribution that seems to have a dip just to one side of the cutoff, with those observations sitting just on the other side, this may indicate manipulation
---
# The density discontinuity test
</br>
**Steps**
</br>
- Estimate the density of the treatment variable. Allow that density to have a discontinuity at the cutoff.
- Look for a significant discontinuity at the cutoff
- Do graphical inspection of the density
<br>
**A big discontinuity is not a good sign. It implies manipulation.**
---
# The density discontinuity test
.vcenter[
.blockquote[
### Example: Manacorda et al. (2011)
</br>
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_c5ea377e283f_files/figure-html/unnamed-chunk-15-1.png" alt="13: Distribution of the binned running variable" width="70%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">13: Distribution of the binned running variable</p>
</div>
<br>
]]
---
#